On some functional calculus of closed operators in a Banach space (Q385258)

Let \([a,b]\) be an interval on the real line. The authors consider the class of functions \(\widetilde Q[a, b]\) consisting of all functions of the form \[ f(z)=\alpha+\beta z+ y\int^b_a {d\tau(t)\over t-z}, \] where \(\tau(t)\) is a bounded non-constant and non-decreasing function, and \(\alpha,\beta,\gamma\) are arbitrary complex numbers. Let now \(X\) be a complex Banach space and let \(A\) be a closed operator on \(X\) whose spectrum does not intersect \([a,b]\), i.e., its resolvent \(R(t, A)= (tI- A)^{-1}\) is defined on \([a,b]\). For any such operator \(A\) and any \(f\in\widetilde Q[a, b]\), the authors define the operator \[ f(A)=\alpha I+\beta A+ \int^b_a R(t, A)\,d\tau(t) \] and show that it is closed. Further, they prove that the map \(f\mapsto f(A)\) is a good functional calculus.